Ask Dr. Math:FAQ

The study of probability helps us figure out the likelihood of something
happening. For instance, when you roll a pair of dice, you might ask how
likely you are to roll a seven. In math, we call the "something happening"
an "event."

The probability of the occurrence of an event can be expressed as a fraction
or a decimal from 0 to 1. Events that are unlikely will have a probability
near 0, and events that are likely to happen have probabilities near 1.*

In any probability problem, it is very important to identify all the
different outcomes that could occur. For instance, in the question about
the dice, you must figure out all the different ways the dice could land,
and all the different ways you could roll a seven.

* Note that when you're dealing with an infinite number of
possible events, an event that could conceivably happen might have
probability zero. Consider the example of picking a random number between 1
and 10 - what is the probability that you'll pick 5.0724? It's zero, but it
could happen.

Likewise, when dealing with infinities, a probability of 1 doesn't
guarantee the event: when choosing a random number between 1 and 10, what is
the probability that you'll choose a number other than 5.0724?
It's 1.

One event, all outcomes equally likely

Suppose we have a jar with 4 red marbles and 6 blue marbles, and we want to
find the probability of drawing a red marble at random. In this case we
know that all outcomes are equally likely: any individual marble has the
same chance of being drawn.

To find a basic probability with all outcomes equally likely, we use a fraction:

number of favorable outcomes
---------------------------------------
total number of possible outcomes

What's a favorable outcome? In our example,
where we want to find the probability of drawing a red marble at random, our
favorable outcome is drawing a red marble.

What's the total number of possible outcomes? The total number of
possible outcomes forms a set called a sample space(see note). In our problem, the sample space consists of
all ten marbles in the jar, because we are equally likely to draw any
one of them.

Using our basic probability fraction, we see that the probability of drawing
a red marble at random is:

number of red marbles

4

---------------------------

=

---

total marbles in jar

10

Since 4/10 reduces to 2/5, the probability of drawing a red marble where all
outcomes are equally likely is 2/5. Expressed as a decimal, 4/10 = .4; as a
percent, 4/10 = 40/100 = 40%.

Suppose we number the marbles 1 to 10. What is the probability of picking out
number 5?

Well, there is only one number 5 marble, and there are still 10 marbles in the jar, so the answer is 1 marble (favorable outcome) divided by 10 marbles (size of sample space) = 1/10 or 10 percent.